Harborth, Heiko Number of odd binomial coefficients. (English) Zbl 0323.10043 Proc. Am. Math. Soc. 62, 19-22 (1977). Summary: Let \( F(n)\) denote the number of odd numbers in the first \(n\) rows of Pascal’s triangle, and \(\theta = (\log 3)/\log 2)\). Then \(\alpha = \lim \sup F(n)/{n^\theta } = 1\), and \( \beta = \lim \inf F(n)/{n^\theta } = 0.812\;556\; \ldots \). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 3 ReviewsCited in 13 Documents MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities 05A10 Factorials, binomial coefficients, combinatorial functions Keywords:Pascal triangle; number of odd numbers; first n rows PDFBibTeX XMLCite \textit{H. Harborth}, Proc. Am. Math. Soc. 62, 19--22 (1977; Zbl 0323.10043) Full Text: DOI Online Encyclopedia of Integer Sequences: Number of entries in n-th row of Pascal’s triangle not divisible by 3. Stolarsky-Harborth constant; lim inf_{n->oo} F(n)/n^theta, where F(n) is the number of odd binomial coefficients in the first n rows and theta=log(3)/log(2). Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth’s recurrence. a(n) = A006046(A077465(n)). Sum of binary digits of A077465(n). Cumulative minima of A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum. a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1. a(0)=1, a(1)=1, a(n)=7*a(n/2) for n=2,4,6,..., a(n)=6*a((n-1)/2)+a((n+1)/2) for n=3,5,7,.... a(0)=1, a(1)=1, a(n) = 17*a(n/2) for n=2,4,6,..., a(n) = 16*a((n-1)/2) + a((n+1)/2) for n=3,5,7,.... a(0)=1, a(1)=1, a(n) = 13*a(n/2) for n=2,4,6,..., a(n) = 12*a((n-1)/2) + a((n+1)/2) for n=3,5,7,.... a(0)=1, a(1)=1, a(n) = 11*a(n/2) for even n, and a(n) = 10*a((n-1)/2) + a((n+1)/2) for odd n >= 3. a(0)=1, a(1)=1, a(n) = 9*a(n/2) for even n >= 2, and a(n) = 8*a((n-1)/2) + a((n+1)/2) for odd n >= 3. a(0)=0, a(1)=1, and for n>=2, a(2*n) = a(n), a(2*n+1) = 2*a(n) + a(n+1). References: [1] N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589 – 592. · Zbl 0030.11102 [2] Heiko Harborth, Über die Teilbarkeit im Pascal-Dreieck, Math.-Phys. Semesterber. 22 (1975), 13 – 21 (German). · Zbl 0296.10009 [3] David Singmaster, Notes on binomial coefficients. III. Any integer divides almost all binomial coefficients, J. London Math. Soc. (2) 8 (1974), 555 – 560. · Zbl 0293.05007 [4] K. B. Stolarsky, Digital sums and binomial coefficients, Notices Amer. Math. Soc. 22 (1975), A-669. Abstract #728-A7. [5] Kenneth B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math. 32 (1977), no. 4, 717 – 730. · Zbl 0355.10012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.